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Let's break down the math problems visible in the image.

### GCF (Greatest Common Factor) Problems:
1. **Find the GCF of 12 and 18:**
   - The factors of 12 are: \( 1, 2, 3, 4, 6, 12 \)
   - The factors of 18 are: \( 1, 2, 3, 6, 9, 18 \)
   - The GCF is **6**, which is the largest common factor between the two numbers.

2. **Find the GCF of 9 and 36:**
   - The factors of 9 are: \( 1, 3, 9 \)
   - The factors of 36 are: \( 1, 2, 3, 4, 6, 9, 12, 18, 36 \)
   - The GCF is **9**.

3. **Find the GCF of 45 and 60:**
   - The factors of 45 are: \( 1, 3, 5, 9, 15, 45 \)
   - The factors of 60 are: \( 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 \)
   - The GCF is **15**.

4. **Find the GCF of 35 and 60:**
   - The factors of 35 are: \( 1, 5, 7, 35 \)
   - The factors of 60 are: \( 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 \)
   - The GCF is **5**.

5. **Find the GCF of 12 and 42:**
   - The factors of 12 are: \( 1, 2, 3, 4, 6, 12 \)
   - The factors of 42 are: \( 1, 2, 3, 6, 7, 14, 21, 42 \)
   - The GCF is **6**.

6. **Find the GCF of 16 and 100:**
   - The factors of 16 are: \( 1, 2, 4, 8, 16 \)
   - The factors of 100 are: \( 1, 2, 4, 5, 10, 20, 25, 50, 100 \)
   - The GCF is **4**.

### Use GCF and the Distributive Property:
7. **20 + 35:**
   - GCF of 20 and 35 is **5**.
   - Expressing it as a product using the distributive property: \( 5(4 + 7) \).

8. **18 + 27:**
   - GCF of 18 and 27 is **9**.
   - Expressing it as a product using the distributive property: \( 9(2 + 3) \).

9. **54 + 45:**
   - GCF of 54 and 45 is **9**.
   - Expressing it as a product using the distributive property: \( 9(6 + 5) \).

### Problem Solving:
10. **Trophies problem:**
    - There are 18 basketball trophies and 24 soccer trophies.
    - The GCF of 18 and 24 is **6**.
    - Therefore, the greatest number of trophies that can go on each shelf is **6**.

11. **Students problem:**
    - There are 24 sixth graders and 40 seventh graders.
    - The GCF of 24 and 40 is **8**.
    - Therefore, the greatest number of students in each group is **8**.

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Would you like a more detailed explanation of any of these solutions?

Here are some related questions:
1. What is the difference between LCM and GCF?
2. Can you explain how to find the GCF using prime factorization?
3. How does the distributive property help with factoring?
4. What is the GCF of three numbers, for example, 12, 18, and 24?
5. How would the GCF change if the number of sixth graders or seventh graders were different?

**Tip:** Always list out the factors of numbers to find the GCF efficiently!

There are 24 sixth graders and 40 seventh graders. Mr. Chan wants to divide both grades into groups of equal size, with the greatest possible number of students in each group. How many students should be in each group?

This problem asks to find the greatest common factor (GCF) of 24 sixth graders and 40 seventh graders to determine the maximum size of equal groups. Suitable for students in grades 5-7, this exercise helps build understanding of GCF and division.

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